Problem 51 - Precalculus : Building Concepts and Connections

Question

This set of exercises will draw on the ideas presented in this section and your general math background. Find \(a\) such that \(\langle 4, a\rangle\) and \langle-3,2\rangle are orthogonal.

Step-by-Step Solution

Answer

The value of \(a\) that makes vectors \(\langle 4, a\rangle\) and \(\langle-3,2\rangle\) orthogonal is \(a = 6\).

Step 1: Understand Dot Product

First, recall the dot product of two vectors \(\langle a, b\rangle\) and \(\langle c, d\rangle\) is calculated by multiplying the corresponding components and adding the products, that is \(ac + bd\).

Step 2: Set Up The Equation

Since \(\langle 4, a\rangle\) and \(\langle-3,2\rangle\) are orthogonal, their dot product should be zero. So, using the calculation for dot product, the equation becomes: \(4*(-3) + a*2 = 0\).

Step 3: Solve The Equation

Now, solve the equation for \(a\). The equation simplifies to \(-12 + 2a = 0\). Adding 12 to both sides of the equation gives \(2a = 12\), and dividing both sides by 2 gives the solution \(a = 6\).

Key Concepts

Dot Product

The dot product, also known as the scalar product, is a fundamental operation in vector calculus. It measures how much two vectors align with each other.
It is particularly useful for determining the angle between two vectors or their orthogonality.
The dot product of two vectors \( \langle a, b \rangle \) and \( \langle c, d \rangle \) is given by the formula:

• Multiply the corresponding components: \( a \cdot c \) and \( b \cdot d \).
• Add these products together: \( ac + bd \).

The result is a scalar, hence the name scalar product. If the vectors are orthogonal (perpendicular), their dot product will be zero.
This is because the cosine of 90 degrees, which represents the angle between orthogonal vectors, is zero.

Orthogonal Vectors

Orthogonality is when two vectors are at a right angle to each other, making them perpendicular.
This property is key in many areas such as physics, engineering, and computer graphics.
For vectors \( \langle a, b \rangle \) and \( \langle c, d \rangle \) to be orthogonal, their dot product must be zero.

• Using the formula \( ac + bd = 0 \), we can determine if two vectors are orthogonal.

This condition highlights how vector components interact.
If we set up a problem so that the product results in zero, we ensure orthogonality between vectors.
Understanding orthogonality helps in simplifying problems and ensuring that directions or forces are independent of each other, which is crucial in mechanics and other fields.

Equation Solving

Equation solving in vector calculus often involves setting up equations that relate to properties like orthogonality or alignment.
In our specific exercise, solving for cross-component values means isolating a variable to satisfy certain vector conditions.
Starting with the equation for orthogonality \( 4 \times (-3) + a \times 2 = 0 \), we simplify this by handling each algebraic step:

• First, multiply the given constants: \( -12 + 2a = 0 \).
• Next, solve for \( a \) by isolating it on one side of the equation: add 12, resulting in \( 2a = 12 \).
• Finally, divide by 2, which gives \( a = 6 \).

These steps highlight the importance of understanding algebraic manipulation within vector contexts.
Each resolved equation offers insight into the precise numerical relationships needed for vectors to meet specified coordinate criteria.